Increasing convex-like function in Hilbert space

Question

I am intersted with the differential equation

$$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two conditions:

1) For all $x,y\in H$ and $t\in \mathbb{R}$ $$<f(t,x)-f(t,y),x-y> \ \geq0.$$

2) If for some $x,y\in H$ and $t\in \mathbb{R}$ $$<f(t,x)-f(t,y),x-y>=0,$$ then $$f(t,x)=f(t,y)=f(t,\frac{x+y}{2}).$$

Answer 1

The choice $f = 0$ satisfies your requirements.

In the case $H = \mathbb{R}$, your conditions are equivalent to $f$ being monotonically increasing.